COMPLEX ANALYSIS GENERAL EXAM FALL 2022

Solve as many problems as you can. Full solutions on a smaller number of problems will be worth more than partial solutions on several problems. Throughout $\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}$. Don’t use any of the Picard theorems.

Problem 1

Compute, for $\xi>0, b>0$ and $a \in \mathbb{R}$

$$\int_{-\infty}^{\infty} \frac{e^{i x \xi}}{(x-a)^{2}+b} d x$$

Show all estimates.

Problem 2

Let $f$ be an entire function with

$$\lim _{R \rightarrow \infty}\left(\sup _{|z|>R} \frac{|f(z)|}{|z|}\right)=0 .$$

Show that $f$ is constant.

Problem 3

Let $R>0$, and $B_{R}(0)=\{z \in \mathbb{C}:|z|<R\}$. Suppose that $f: \mathbb{C} \backslash \overline{B_{R}(0)} \rightarrow \mathbb{C}$ is analytic and that $\lim _{z \rightarrow \infty} f(z)=0$. Show that $\lim _{z \rightarrow \infty} z f(z)$ exists.

Hint: it maybe to helpful to consider the "singularity of $f$ at $\infty$ " namely, the signularity of $g(z)=f(1 / z)$ at $z=0$.

Problem 4

Suppose that $f_{n}$ is a sequence of entire functions and that $f_{n}$ converges uniformly on compact subsets of $\mathbb{C}$ to a polynomial $p$ of degree $d$. Show that there is an $N \in \mathbb{N}$ so that for all $n \geq N$, the function $f_{n}$ has at least $d$ zeroes (counted with multiplicity).

Problem 5

Let $U$ be open and connected. Recall that a collection $\mathcal{F}$ of analytic functions $U \rightarrow \mathbb{C}$ is normal if given any sequence $\left(f_{n}\right)_{n}$ in $\mathcal{F}$, there is a subsequence $f_{n_{k}}$ which converges uniformly on compact sets. Let $\mathbb{H}=\{z \in \mathbb{C}: \operatorname{Im}(z)>0\}$. Fix $p \in U$ and let $\mathcal{F}$ be the collection of analytic functions $f: U \rightarrow \mathbb{H}$ so that $f(p)=i$. Show that $\mathcal{F}$ is normal.